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Estimation of quantile oriented sensitivity indices
Véronique Maume-Deschamps, Ibrahima Niang
To cite this version:
ESTIMATION OF QUANTILE ORIENTED SENSITIVITY INDICES
V ´ERONIQUE MAUME-DESCHAMPS AND IBRAHIMA NIANG
Abstract. The paper concerns quantile oriented sensitivity analysis. We rewrite the corresponding indices using the Conditional Tail Expec-tation risk measure. Then, we use this new expression to built estima-tors.
1. Introduction
Many models encountered in applied sciences involve input parameters which are often not precisely known. Some of the input variables may strongly affect the output, while others have a small effect. Sensitivity anal-ysis aims at measuring the impact of each input parameter uncertainty on the model output and, more specifically, to identify the most sensitive pa-rameters (or groups of papa-rameters). One of the common metrics to evaluate the sensitivity is the Sobol index. More precisely, given two random vari-ables X and Y , the X-Sobol index on Y compares the total variance of Y to the expected variance of the variable Y conditioned by X, i.e:
(1.1) SX = Var(E(Y | X))
Var(Y ) .
It rewrites SX = 1−E((VarYVar(Y )| X)). It is a statistical measure of the
rela-tive impact of X on the variability of Y ; the most sensirela-tive parameters can be identified and ranked as the parameters with the largest Sobol indices. Introduced in [7], contrast indices named ”Goal Oriented Sensitivity Analy-sis” generalize the Sobol ones. The construction of these indices is based on contrast functions. Roughly speaking, given ψ a contrast function and X, Y two random variables, the ψ-X contrast index on Y is given by (see [7]):
SXψ = min
θ∈R(E[ψ(Y ; θ)])− E(minθ∈R[E(ψ(Y ; θ)|X)])
min θ∈R(E[ψ(Y ; θ)])− E min θ∈Rψ(Y ; θ) .
SψX represents a statistical indicator of the impact of X on the variability of the output function Y with respect to the argmin of a contrast function. If we consider the mean-constrat function ψ : (y, θ) 7→ (y − θ)2, then we
retrieve the first order Sobol indices defined in (1.1) and it measures the impact of X to the deviation of Y with respect to the mean. There exists a pretty large literature in the case where the contrast function is given by the mean-constrat function ψ : (y, θ) 7→ (y − θ)2 (see for example [15], [2],
Key words and phrases. Quantile oriented sensitivity analysis.
[3], [14] or [9]).
In this paper, we focus on contrast indices obtained with the α-quantile contrast function, i.e., with the contrast function given by:
(1.2) ψα: (y, θ)7→ (y − θ)(α − 1y≤θ), α∈]0, 1[
We call quantile contrast index the sensitivity measure that one obtain with the contrast function given by (1.2). This setting is also called quantile
oriented sensitivity analysis (QOSA) (see [5]). In this work, we show that
quantile contrast indices can be linked with the Conditional Tail Expecta-tion (or CTE) risk measure. Then, from this expression, we propose an estimation method based on Monte Carlo sampling techniques. Note that this approach is the first formal estimation of the quantiles oriented sensi-tivity indices. It appears firstly in [12]. Nevertheless, during the writing of this article, we have been informed of a recent work ([5]) where another procedure is proposed.
The paper is organized as follows. In Section 2, we recall basic definitions on the CTE and the contrast indices. In Section 3, we show that quantile contrast may be rewritten in terms of the CTE. Finally, Section 4, is devoted to the estimation of quantile contrast indices and to some simulations.
2. Main definitions
Let us consider a measurable function f and a random vector X = (X1, . . . , Xk) ∈ Rk, k ≥ 1. Let Y be the real-valued response variable:
Y = f (X). We assume that the random input variables Xi, i≥ 1 are
inde-pendent, which is the standard setting of sensitivity analysis.
For a random variable Z, FZ denotes its distribution function, and FZ−1 the
generalized inverse of FZ (or the quantile function). We recall that for a
continuous random variable Z with moment of order 1, the Conditional Tail Expectation denoted CTE of Z at level α∈]0, 1[ is given by:
(2.1) CT Eα(Z) = E Z|Z > FZ−1(α)= 1 1− α Z 1 α FZ−1(u)du.
From an actuarial point of view, the CTE measures the average of losses given that a specified confidence level α is exceed. We refer the reader to [6] for a detailed review. In what follows, we introduce some concepts about contrast functions and contrast index. The reader is referred for instance to [7]. We assume that the output Y = f (X) is a continuous random variable. Given a function ψ
ψ : R× R → R+ (y, θ) 7→ ψ(y, θ),
such that there is a unique minimum θ∗ ∈ R to E[ψ(Y ; θ)]. Some examples of contrast functions that allow to estimate various parameters associated to a probability distribution are listed on Table 1.
Contrast functions θ∗
The mean ψ(y, θ) =|y − θ|2 θ∗= E(Y )
The median ψ(y, θ) =|y − θ| θ∗= FY−1(12) The α-quantile ψ(y, θ) = (y− θ)(α − 1y≤θ) θ∗= FY−1(α)
Table 1. Contrast functions exemples
In what follows, we focus on α- quantile contrast function, i.e the contrast function given by:
(2.2) ψα: (y, θ)7→ (y − θ)(α − 1y≤θ), α∈]0, 1[
The quantile oriented index associated to the input Xi is:
(2.3) SαX
i =
min
θ∈R(E[ψα(Y ; θ)])− E(minθ∈RE[ψα(Y ; θ)|Xi])
min
θ∈R(E[ψα(Y ; θ)])
Remark. It is straightforward to see that if Y and Xi are independent then
SαX
i = 0 and if Y is Xi-measurable then S
α
Xi = 1, which is an expected
behavior for sensitivity indices.
We are interested on how one can express the quantile contrast index in terms of the Conditional Tail Expectation (CTE) risk measure.
3. Relation between contrast indices and Conditional Tail Expectation (CTE) risk measure
We recall that, for X and Y two random variables, the conditional cu-mulative distribution FY|X and the conditional quantile FY−1|X at level α are defined respectively as:
FY|X(y) = P (Y ≤ y|X) , y ∈ R. (3.1) FY−1|X(α) = infny∈ R| FY|X(y)≥ αo.
These are X-measurable random functions. In the following proposition, we give a relation between the risk measure CTE and quantile contrast index. Proposition 3.1. Assume that E|Y | < ∞, then ∀α ∈]0, 1[, ∀i = 1, . . . , k, (3.2) SXαi = 1−E Y|Y > FY−1|X i(α) − E(Y ) CT Eα(Y )− E(Y ) .
Proof. SXαi rewrites as:
Consider θ∗
1 = argmin θ∈R E[ψα
(Y ; θ)]. We have θ∗
1 = F−1(α).
On one other side, E(min θ∈R[E(ψα(Y ; θ)|Xi)]) = E (E (ψα(Y ; θ ∗ 2)|Xi)) = E (ψα(Y ; θ∗2)) , whit θ∗ 2 = argmin θ∈R E [ψα
(Y ; θ)|Xi] it is a Xi measurable random variable. Let
Z = Y − FY−1|X
i(α).
Remark that, ∂E[ψα(Y ;θ)|Xi]
∂θ =−α + P(Y ≤ θ|Xi) and θ∗2 = FY−1|Xi(α). We deduce: E (ψα(Y ; θ2∗)) = E Z1Y >F−1 Y|Xi(α) − (1 − α)E(Z)
= P(Y > θ∗2)E(Y|Y > θ∗2)− P(Y > θ∗2)E(θ∗2|Y > θ2∗)− (1 − α)E(Z).
In addition,
P(Y > θ∗2) = 1−P(Y ≤ θ2∗) = 1−E E(1Y≤θ∗ 2|Xi)
= 1−E FY|Xi(θ2∗)= 1−α. On one other side, conditioning by Xi, leads to
E(θ∗2|Y > θ∗2) = 1 1− αE θ ∗ 21Y >θ∗ 2 = 1 1− αE(E θ ∗ 21Y >θ∗ 2|Xi ) = E(θ∗2). Consequently,
E (ψα(Y ; θ2∗)) = (1− α)E(Y |Y > θ∗2)− (1 − α)E(θ∗2)− (1 − α)E(Y ) + (1 − α)E(θ2∗)
= (1− α) (E(Y |Y > θ∗2)− E(Y )) .
which concludes the proof.
We shall use this expression of the quantile oriented indices to propose an estimation of them.
4. Estimation of quantile contrast index
We consider the statistical estimation of quantile contrast index. Fix an index i = 1, . . . , k.We assume that E|Y | < ∞. We consider, for α ∈]0, 1[. We have (4.1) SXαi = 1−E Y|Y > FY−1|X i(α) − E(Y ) CT Eα(Y )− E(Y ) .
4.1. Estimation procedure. Let θ∗ = F−1
Y (α) and θi(x) = FY−1|Xi=x(α) be
the respective quantile at level α of Y and of Y|Xi= x.
Consider Y1, . . . , Yn are n, n≥ 1 an i.i.d. n-sample of the distribution of Y .
Fn denotes the empirical distribution function:
Fn(y) = 1 n n X i=1 1{Yi≤y},∀y ∈ R and bθ∗ is the classical quantile estimator.
The conditional distribution function FY|Xi=xcan be estimated by a Kernel
estimator: Fn(y|Xi= x) = Pn j=1K x−Xij hn 1{Yj≤y} Pn j=1K x−Xij hn ,
where K is a kernel and hn is a positive number depending on the sample
size n, called bandwidth. Then θi(x) is obtained by
(4.2) θˆi(x) = Fn−1(α|x) = inf {y : Fn(y|x) ≥ α} ,
We refer the reader to [1]; [11]; [8] or [16] for a detailed review on quantile and conditional quantile estimation.
In what follows, we give an estimation procedure for estimating quan-tile contrast index. It requires the two following steps (recall that Y = f(X1, . . . , Xk))
(1) Generate X1j, . . . , Xkj and compute the Yj = f (X1j, . . . , Xkj), for j =
1, . . . , n. Replace in (4.1) the expectation E(Y ) by its empirical ver-sion.
(2) Generate X∗j = (X∗j
1 , . . . , Xk∗j) for j = 1, . . . , n (independent copies
of the previous vectors) and compute the Yj∗ = f (X1∗j, . . . , Xk∗j), j = 1, . . . , n. Then from the sample Y∗
j = f (X1∗j, . . . , Xk∗j), j = 1, . . . , n
compute ˆθ∗ and ˆθi(Xi).
An estimator of the index (4.1) is given by (4.3) Sn,Xα i = 1− 1 n(1−α) Pn j=1Yj1Yj> ˆθi(Xij)− 1 n Pn j=1Yj 1 n(1−α) Pn j=1Yj1Y j> ˆθ∗− 1 n Pn j=1Yj . Remark that equation (4.3) is equivalent to:
conditionally to the sample X∗, are two independent and identically
dis-tributed (i.i.d) sample of the distribution of Ri and Z where
Ri = Y 1 1− α1Y >F−1 Y|Xi(α)− 1 , Z = Y 1 1− α1Y >FY−1(α)− 1 . For the sake of simplicity, we denote by
¯ Zn= 1 n n X j=1 Zj, R¯n= 1 n n X j=1 Rij.
Proposition 4.1. (Consistency). Assume that E|Y | < ∞, then
(4.5) Sn,Xα i a.s −→ n→∞S α Xi
Proof. The result is a straightforward application of the strong law of large
number conditionally to X∗.
Proposition 4.2. (Asymptotic normality). Assume that Y is square inte-grable. Then, (4.6) √n(Sn,Xα i− S α Xi) L −→ n→∞N 0, σ 2 S , where σS2 = Var(R i)− 2β(1 − Sα Xi) + (1− S α Xi) 2Var(Z) (CT Eα(Y )− E(Y ))2 , and β = Cov(Ri, Z).
Proof. We do the proof conditionally to X∗. Denote
Sn,Xα
i = h( ¯Un),
where Uj = (Rij, Zj)T and
h(x, y) = 1−xy. The central limit theorem gives that:
√
n( ¯Un− µ) −→L
n→∞N2(0, Γ) ,
where Γ is the covariance matrice of (Ri, Z) and µ =
E(Ri)
E(Z)
. Since the function h is differentiable at µ, the Delta method gives:
√ n(Sn,Xα i− SXαi) −→L n→∞N 0, g TΓg, where g=∇h(µ).
By differentiation, we get that, for any x, y so that y 6= 0: ∇h(x, y) = −1y, x y2 T . Hence, using 4.1, we get that
Thus gTΓg = Var(Ri)(1− S α Xi) 2 E(Ri)2 − 2Cov(R i, Z)(1− S α Xi) 3 E(Ri)2 + Var(Z) (1− Sα Xi) 4 E(Ri)2 = (1− S α Xi) 2 E(Ri)2 Var(R i)− 2Cov(Ri, Z)(1− Sα Xi) + Var(Z)(1− S α Xi) 2. Remark that (1−S α Xi) 2 E(Ri)2 = 1
E(Z)2 and that E(Z) = CT Eα(Y )− E(Y ), hence
we deduce that gTΓg = Var(R i)− 2β(1 − Sψα Xi) + (1− S α Xi) 2Var(Z) (CT Eα(Y )− E(Y ))2 ,
where β = Cov(Ri, Z) which concludes the proof.
Remark. When computing σS2, we replace the variance of Ri and Z as well
as the covariance of the random vector (Ri, Z) by their empirical versions.
Remark. Following [10], in order to improve the efficiency of the estimator,
CT Eα(Y ) and E(Y ) could be estimated by using the complete sample X, X∗.
A further study is needed to get its asymptotic properties.
4.2. Numerical illustrations. In order to validate our estimation proce-dure, we illustrate the asymptotic results of Proposition 4.2 in the following example that is considered in [7].
Example 1. Let us considerer an output of type Y = f (X1, X2) = X1+ X2.
where X1 ∼ Exp(1) and X2 ∼ −X1 with X1 and X2 independent. The
quantile contrast index of X1 and X2 are known analytically (see [7]) and
they are given by
SXα1 = (1− α)(1 − log(2(1 − α))) + α log(α) (1− α)(1 − log(2(1 − α))) if α≥ 1 2 α(1− log(2α)) + α log(α) α(1− log(2α)) if α < 1 2 and SXα2 = (1− α)(1 − log(2(1 − α))) + (1 − α) log(1 − α) (1− α)(1 − log(2(1 − α))) if α≥ 1 2 α(1− log(2α)) + (1 − α) log(1 − α) α(1− log(2α)) if α < 1 2 In the case where α = 12, then Sα
X1 = S
α X2.
Table 2 displays the sensitivity of X1 and X2 for different values of α. We
estimate for each variable a 95% confidence interval. We denote by SX1
n,α and
SX2
n,α as the empirical estimator of SXα1 and S
α
X2 given in (4.3) for a sample
size n = 100000. The conditional quantile of Y|X1 and Y|X2 are computed
using a gaussian kernel with a bandwidth hn = n−
1
5. The quantity IC
1
and IC2 denote the respective confidence interval of X1 and X2. Table 2
X1and X2which measures the relative average of the square of the ”errors”,
that is the following quantity RM SEXi = v u u t 1 n n X j=1 (SXi n,α)j − SXαi SXα i !2 , i= 1, 2.
As we can see in Table 3, the RMSE is low which assess the reliability of our estimator. From this result, we can conclude that the estimated indices SXi
n,α, i= 1, 2 is near to the true values modulo the Monte- Carlo errors due
to the numerical simulation of the indices.
α SαX1 Sn,Xα 1 IC1 SXα2 S α n,X2 IC2 0.05 0.0929 0.0950 [0.0807, 0.1052] 0.7049 0.7005 [06917, 0.7181] 0.1 0.1176 0.1166 [0.1097, 0.1255] 0.6366 0.6397 [0.6259, 0.6474] 0.5 0.3069 0.3108 [0.3009, 0.3128] 0.3069 0.3042 [0.3010, 0.3127] 0.7 0.4491 0.4436 [0.4417, 0.4566] 0.2031 0.2026 [0.1980, 0.2082] 0.99 0.7974 0.7907 [0.7756, 0.8193] 0.0625 0.0630 [0.0309, 0.0941]
Table 2. Quantile contrast index confidence interval at level 95% for different values of α.
α SXα1 RM SE SXα2 RM SE 0.05 0.0929 0.0318 0.7049 0.0064
0.1 0.1176 0.0241 0.6366 0.0065 0.5 0.3069 0.0091 0.3069 0.0096 0.7 0.4491 0.0075 0.2031 0.0130 Table 3. Relative mean square error (RMSE).
Example 2. (Vasicek Model) Here we present a financial application of the use of quantile contrast index. We focus on the classical Vasicek model where the yield curve is given as an output of an instantaneous spot rate model with the following risk-neutral dynamics
(4.7) drt= a(b− rt)dt + σdWt
where A(t, T ) = exp (b− σ 2 2a2)(B(t, T )− (T − t)) − σ2 4aB 2(t, T ) and B(t, T ) = 1− e −a(T −t) a .
The yield-curve can be obtained as a deterministic transformation of zero-coupon bond prices at different maturities.
In what follows, we quantify the sensitivity of the input parameters{a, b, σ} affecting the uncertainty in the bond price at time t = 0. In the following numerical experiments, the maturity T and the initial spot rate r0are chosen
such that T = 1 and r0 = 10%. Table 4 reports the estimated quantile
contrast indices of the parameter a, b, σ for different values of α where the probability laws of these parameters are uniform on [0, 1]. For this model, no closed form formula is available. Contrary to Sobol indices, the sensitivity
α Sbaα Sbbα Sbσα 0.05 0.4961 0.5961 0.0210 0.1 0.1948 0.4036 0.0667 0.5 0.1722 0.2685 0.0508 0.7 0.1096 0.1679 0.1913 0.9 0.1053 0.0025 0.2928 0.99 0.4744 0.2596 0.3965
Table 4. Estimation of α-quantile contrast indices for dif-ferent values of α
of our model parameters strongly depend on the confidence level α. For α ∈ {0.05, 0.1, 0.5, }, most of the uncertainty in the bond price is due to the long-term mean-reversion level b and the speed of mean reversion a whereas for α = 0.9 or α = 0.99, we can see that a and σ becomes more important than the the long-term mean-reversion level b in terms of output uncertainty.
5. Conclusion
The main goal of this paper was to propose an estimation of the quantile oriented sensitivity indices. Our proposition if based on a rewriting of these indices using the Conditional Tail Expectation. This domain deserve much additional work in order to make the QOSA a useful and practical tool. In particular, their performance with respect to the procedure proposed in [5] has to be studied.
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